Mourinho, who claimed last week that everything he says and does is "mind games", returned fire by referring to Arsenal's lack of silverware since they won the FA Cup in 2005.
"He is a specialist in failure, I'm not," Mourinho told a news conference.
"If supposedly he (Arsene Wenger) is right and I am afraid of failure it is because I don't fail many times.
"He is specialist because eight years without silverware is very long. If I do that with Chelsea, I leave and I don't come back."
The remarks recalled Mourinho's famous outburst against Wenger in 2005 when he labelled the Arsenal manager a "voyeur", after becoming annoyed at comments from the Frenchman about his Chelsea team.
Chelsea, who face Manchester City in the fifth round of the FA Cup on Saturday, are top of the Premier League, a point clear of Arsenal in second.
After each manager labelled the other team as being title favourites by virtue of how much they had spent in the transfer market, Mourinho hinted City were not operating within the same UEFA Financial Fair Play guidelines.
He then turned his attentions to City's Ivory Coast midfielder Yaya Toure, who aimed an off-the-ball kick at Norwich striker Ricky van Wolfswinkel in their 0-0 stalemate last weekend but escaped any FA punishment.
Chilean coach Pellegrini has been reluctant to involve himself in a verbal spat and said this week that mental strength will decide who wins the Premier League and not mind games.
"I don't like this history of records between managers because it is between teams," said Mourinho, who will be without injured defenders Gary Cahill and John Terry for the trip to the Etihad.
"If he (Pellegrini) doesn't like to speak about me perfect, I hope it stays that way."
Heavyweights City and Chelsea meet at the Etihad Stadium for their second clash in quick succession, after the London club won 1-0 to complete a league double over their title rivals 12 days ago.
"It is not easy to play consecutive matches against the same team. You try to read the previous game and try to improve. Sometimes you can be influenced by the direction the game takes," Mourinho added.
"We know they are strong. Our priority is always the next match."